A New Accurate and Efficient Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations: Approach Based on Geometric Considerations

This paper deals with a new numerical iterativemethod for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphsonmethod. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.


Introduction
Solving both nonlinear equations and systems is a situation very often encountered in various fields of formal or physical sciences. For instance, solid mechanics is a branch of physics where the resolution of problems governed by nonlinear equations and systems occurs frequently [1][2][3][4][5][6][7][8][9][10]. In most cases, Newton method (also known as Newton-Raphson algorithm) is most commonly used for approximating the solutions of scalar and vector nonlinear equations [11][12][13]. But, over the years, several other numerical methods have been developed for providing iteratively the approximate solutions associated with nonlinear equations and/or systems [14][15][16][17][18][19][20][21][22][23][24][25]. Some of them present the advantage of having both high accuracy and strong efficiency using a numerical procedure based on an enhanced Newton-Raphson algorithm [26]. In this study, we propose to improve the iterative procedure developed in previous works [27,28] for finding numerically the solution of both scalar and vector nonlinear equations. This study is decomposed as follows: (i) in Section 2, a new numerical geometry-based root-finding algorithm coupled with a stationary-type iterative method (such as Jacobi or Gauss-Seidel) is presented with the aim of solving any system of nonlinear equations [29,30]; (ii) in Section 3, the numerical predictive abilities associated with the proposed iterative method are tested on some examples and also compared with other algorithms [31,32].

Used Root-Finding Algorithm (RFA).
In previous works [27,28], a root-finding algorithm (RFA) has been developed for approximating the solutions of scalar nonlinear equations.
The new RFA presented here is an extended version to that previously developed taking into account some geometric considerations. In this paper, we propose to use a RFA coupled with Jacobi and Gauss-Seidel type procedures for iteratively solving nonlinear system (S nl ). Hence, we adopt a new RFA for finding approximate solution +1 (when is fixed and with the known set Δ ) associated with each nonlinear equation of system (S nl ) (see Section 2.1). For each nonlinear equation , parametrized by the set of variables Δ , depending only on one variable +1 , we introduce the exact and inexact local curvature associated with the curve representing the nonlinear equation in question.
(ii) In the second step, we introduce the iterative exact and inexact local curvature associated with the curve representing nonlinear function at point = ( , F ) (see Figure 1): where | * | denotes the absolute-value function associated with the variable * , † R = † ( ; Δ ) is the exact ( † = ex) or inexact ( † = in) radius of the osculating † C at point , † (with † = ex, in) is functional associated with † R , and F = ( ; Δ ) is the second-order derivative of function at point . It should be noted that: (i) we consider the exact radius ex R associated with the true osculating circle ex C at point = ( , F ) (see [33]); (ii) in line with [27,28], we consider an inexact radius associated with the osculating circle in C at point = ( , F ), that is, ex R ̸ = in R (see (7)).
where is a functional associated with W .
(vi) In the sixth step, we introduce the iterative straight line H = ( ; Ξ ) passing through the point ( , F ) and the iterative perpendicular straight line G such as (with † = ex, in) where is a functional associated with H .

Preliminary Remarks.
In this section, we propose to evaluate the predictive abilities associated with the numerical iterative method developed in Section 2.3 (i.e., AGA) on some examples both in the case of scalar and vector nonlinear equations. Hence, AGA is compared to other iterative Newton-Raphson type methods [27,28,[30][31][32] coupled with Jacobi (J) and Gauss-Seidel (GS) techniques. All the numerical implementations associated with these presented iterative methods have been made in Matlab software (see [26,[34][35][36][37][38][39]).
(ii) Standard Newton's Algorithm (SNA): (iii) Third-order Modified Newton Method (TMNM): In order to stop the iterative process associated with each considered algorithm, we consider three coupled types of criteria for dealing with nonlinear equations: where max denotes the maximum number of iterations associated with scalar-valued equations. (b) (C2S) on the residue error, where re is the tolerance parameter associated with the residue error criterion for scalar-valued equations and ‖ †‖ = | †| is the absolute-value norm. (c) (C3S) on the approximation error, where ae is the tolerance parameter associated with the absolute error criterion for scalarvalued equations.
(ii) For vector-valued equations (S nl ): (a) (C1V) on the iteration number, we adopt the same condition that (C1S), that is, ≤ V max = max (where V max denotes the maximum number of iterations associated with vector-valued equations).
(b) (C2V) on the residue error, where V1 re (resp., V2 re ) is the tolerance parameter associated with the residue error criterion for vector-valued equations and ‖Υ‖ = ( . It is important to point out that where ae is the tolerance parameter associated with the absolute error criterion for vectorvalued equations. Here, for the stopping criteria (C1S, C1V), (C2S, C2V), and (C3S, C3V) associated with the iterative process, we consider: (i) the maximum number of iterations max = V max = 20; (ii) the tolerance parameter re = ae = 10 −30 for the scalar-valued equations; (iii) the tolerance parameter V re = V ae = 10 −30 (with = 1,2) for the vector-valued equations.

Examples.
We consider the following nonlinear equations.

Concluding Comments
The present work concerns a new numerical iterative method for approximating the solutions of both scalar and vector nonlinear equations. Based on an iterative procedure previously developed in a study, we propose here an extended form of this numerical algorithm including the use of a stationarytype iterative procedure in order to solve systems of nonlinear equations. A predictive numerical analysis associated with this proposed method for providing a more accurate approximate solution in regard to nonlinear equations and                        systems is tested, assessed and discussed on some specific examples.